Linear group precoding for massive MIMO systems under exponential spatial correlation

Journal of Science and Technique - N.205 (3-2020) - Le Quy Don Technical University 56 LINEAR GROUP PRECODING FOR MASSIVE MIMO SYSTEMS UNDER EXPONENTIAL SPATIAL CORRELATION Dinh Van Khoi1*, Le Minh Tuan2, Ngo Vu Duc3, Ta Chi Hieu1 1Le Quy Don Technical University; 2MobiFone R&D Center, MobiFone Corporation; 3Hanoi University of Science and Technology Abstract In this paper, a low-complexity linear group precoding algorithm in the exponential correlation channel model is proposed for massive MIMO systems. The proposed precoder consists of two components: The first one minimizes the interferences among neighboring user groups; The second one improves the system performance by utilizing the ELR-SLB technique. Numerical and simulation results show that the proposed precoder has remarkably lower computational complexity than its LC-RBD-LR-ZF counterpart, while its bit error rate (BER) performance is asymptotic to that of the LC-RBD-LR-ZF precoder as the number of groups increases. Keywords: MU-MIMO system; massive MIMO system; linear precoding algorithms; nonlinear precoding algorithms; lattice reduction algorithm in MIMO system. 1. Introduction Multiple-Input Multiple-Output (MIMO) technology has been widely studied for years and already implemented in 4G mobile communication systems [1]. The initial research focuses on point-to-point MIMO systems. In recent years, more and more researchers are interested in Multiuser MIMO (MU-MIMO) scenarios. However, a limitation of MU-MIMO system is that BS is usually equipped with small numbers of antenna elements (normally fewer than 10) [2]. Therefore, the spectrum efficiency and system capacity are still relatively modest. To solve this problem, massive MIMO systems have recently been proposed [1, 3, 4, 5]. In the Massive MIMO, the number of antennas at the base station (BS) can be up to hundreds of antennas (or even thousands) to simultaneously serve dozens of users using the same frequency resource. The Massive MIMO system can significantly improve the channel capacity, enhance the spectrum utilization efficiency and quality of the system [4]. It is expected that Massive MIMO will be a key and a potential candidate for the next generation wireless network (e.g., 5G network) [1, 4, 6]. Although the massive MIMO systems have numerous advantages, they face a number of challenges such as hardware complexity, power consumption, and system cost due to the large number of antennas equipped at the BS. Therefore, reducing the * Email: dinhvankhoi.tcu@gmail.com Journal of Science and Technique - N.205 (3-2020) - Le Quy Don Technical University 57 complexity of the signal processing algorithms for both uplink and downlink in massive MIMO systems is essential. In massive MIMO systems, the dimensions of transmit/receive signal vectors are normally very large due to large numbers of antennas and users. Therefore, the precoding algorithms with low-complexity, e.g. Zero Forcing (ZF), Minimum Mean Square Error (MMSE) and Maximum Ratio Transmission (MRT) are considered as suitable solutions [7, 8, 9]. The Dirty Paper Coding (DPC) proposed in [10] can achieve the capacity region for multiuser precoding. However, its complexity becomes significantly large as the system dimensions grow due to the implementation of random nonlinear encoding and decoding [11, 12]. The combination of lattice reduction algorithms and precoding techniques for the downlinks of massive MIMO systems is an important solution to improve the system performance. In [13], the authors adopted the Seysen’s lattice reduction algorithm (SA) to create a LR-aided precoding technique for the MU-MIMO system. The simulation results show that the proposed algorithm gives better performance than the precoding algorithm adopting the Lenstra-Lenstra-Lovasz (LLL) method. In [14], a Block Diagonalization (BD) aided precoding algorithm was proposed based on the Pseudo- Inverse Block Diagonalization (PINVBD) presented in [15] and the QR decomposition of the channel matrix. Furthermore, in each block, the Lattice Reduction and Tomlinson-Halashima precoder (THP) algorithms are applied to improve the quality of the system. In [16], the authors proposed the low-complexity Lattice Reduction (LR)- aided BD algorithms for the MU-MIMO, referred to as LC-RBD-LRZF and LC-RBD- LR-MMSE. In the authors’ proposal, the first precoding matrix is obtained using the QR decomposition instead of the Singular Value Decomposition (SVD). The second precoding matrix is computed based on either the ZF or MMSE algorithm to provide the corresponding LC-RBD-LR-ZF or LC-RBD-LR-MMSE precoder. It was shown in [16] that the precoders significantly improved the system performance, while reducing the computational complexity compared to the original BD one. However, the computational complexities of the precoders presented in [14] and [16] are still very high due to the adoptions of the QR decomposition and THP algorithms. In this paper, we propose a low complexity precoding algorithm for massive MIMO systems using the exponential correlation channel model. Based on the linear precoding algorithms and the lattice reduction technique, we propose the Zero Forcing group precoder combining with the low-complexity lattice reduction technique (or ZF- GP-LR precoder for short). In our proposal, the channel matrix from the BS to all users is divided into L groups (i.e., sub-matrices), each of which contains a number of rows of the channel matrix. The sizes of the sub-matrices are all the same. The proposed Journal of Science and Technique - N.205 (3-2020) - Le Quy Don Technical University 58 precoding matrix is designed to have two components: The first one minimizes the interferences from neighboring user groups by using QR decomposition of the sub- matrices; The second one enhances the system performance thanks to the combination of the Zero Forcing precoding and the ELR-SLB lattice reduction algorithms. Numerical and simulation results show that the ZF-GP-LR precoder has remarkably lower computational complexity than the LC-RBD-LR-ZF in [16], whereas its BER performance is asymptotic to that of the LC-RBD-LR-ZF as L increases. Besides, the complexity of the proposed algorithm grows proportionally to the number of groups. Simulation results also show that the spatial correlation adversely affects the system performance no matter which precoder is adopted. Fortunately, the proposed precoder still works well as compared with the LC-RBD-LR-ZF under such circumstances. The rest of this paper is organized as follows. In Section 2, we present massive MIMO system model. The LC-RBD-LR-ZF and element-based lattice reduction (ELR) algorithms are reviewed in Section 3. The linear group precoding algorithm in combination with ELR-SLB technique is presented in Section 4. Simulation results are evaluated in Section 5. Finally, conclusions are drawn in Section 6. Notation: The notations are defined as follows: Matrices and vectors are represented by symbols in bold; (.)T and (.)H denote the transpose and conjugate transpose, respectively. We denote | |a for the absolute value of scalar a and det(B) for the determinant of B. rounds the real and imaginary parts of the complex number  to the nearest integers. {.}Tr is the trace of a square matrix. 2. The downlink channel model in massive MIMO system Let us consider a massive MIMO system, where the BS is equipped with TN antennas to simultaneously serve K users, each user has uN antennas. Thus, the total number of antennas of K users is R uN KN . In addition, the Channel State Information (CSI) is assumed to be perfectly known at the BS. In reality, although the theoretical distance is guaranteed, there till exist certain amounts of correlation among the antennas. These correlation can be modeled based on the actual measurements. Therefore, spatial correlations always exist among transmit and receive antennas, thereby degrading the system performance. In order to take into account the effect of the spatial correlation, the channel model is given by the following equation [17]: 1/2 1/2 ,H R H Rcorr R T (1) where 1 2 ( ) ( ) ...( ) R T K T N NT T T corr corr corr corr    H H H H  is the channel matrix with antenna correlations, TR is the T TN N transmit correlation matrix and RR is the Journal of Science and Technique - N.205 (3-2020) - Le Quy Don Technical University 59 R RN N receive correlation matrix. H is the uncorrelated channel matrix, whose entries, ijh  , are complex Gaussian random variables with zero mean and unit variance. In this paper, we investigate the massive MIMO system in correlated channels using the exponential correlation matrix model [18]. In this model, the components of TR and RR are determined as follows: * , , | | 1, , v s sv vs r v r r r v s s      (2) where r ≥ 0 is the correlation coefficient between any two neighboring transmit or receive antennas. Let 1uNu x  represents the transmitted signal vector of the uth user. The received signal vector for the uth user, (u = 1, 2,, K), 1uNu y  is given by , , , , , , 1 1, K K corr u corr u k u corr u corr u u corr u corr u k u k k k u       y H W x n H W x H W x n (3) where , u T N N corr u H  is channel matrix from the BS to the uth user; , T uN Ncorr u W  denotes the precoding matrix for the uth user; 1uNu n  is noise vector at the uth user. Note that, in (3), , ,corr u corr u uH W x is the desired signal component of the uth user, , , 1, K corr u corr k k k k u   H W x represents unwanted signals at the uth user. Let 11 1 R T NT T T K    y y y y  be the overall received signal vector for all users. Then, the relationship between the transmitted signal vector, 1RN x  and the received signal vector y can be expressed as ( ),corr corr y H W x n (4) where corrH is channel matrix from BS to all K users, defined in (1); T RN Ncorr W  is the precoding matrix for all users; 1RN n  is noise vector at the K users, whose entries are assumed to be identical independent distributed (i.i.d) random variables with zero mean and variance 2n . 3. Review of LC-RBD-LR-ZF and element-base lattice reduction (ELR) algorithms A. LC-RBD-LR-ZF algorithm The LC-RBD-LR-ZF algorithm is proposed for Multiuser MIMO (MU-MIMO) system using the uncorrelated channel model [16]. This means that the channel matrix from BS to all users is 1 2 ( ) ( ) ...( ) R T K T N NT T T        H H H H  . The precoding matrix Journal of Science and Technique - N.205 (3-2020) - Le Quy Don Technical University 60 of the LC-RBD-LR-ZF algorithm is expressed as follows: ,a bW W W (5) where 1 2, ,..., ;T T N KNa a a a a K u    W W W W W is the precoding matrix for the uth user, created by applying QR decomposition to the channel matrix ,{ } u uuN  H I H ; the matrix 1 1 1 ( ) ...( ) ( ) ...( ) u u u K TT T T T          H H H H H  is obtained by removing ( ) u T H from H ; 2 R n s N E    ; u R uN N N  ; and sE is the energy of each transmitted signal symbol. The QR decomposition of u H is given by . u u u H Q R (6) Then, the precoding matrix auW for the uth user is obtained as ( 1: , 1: ).au u u u T u u TN N N N N N    W Q (7) After getting auW , the effective channel matrix for the uth user is expressed as ˆ , u u a u H H W (8) which is subsequently converted into the LR domain by using the LLL algorithm in [19] as ˆ ,ˆ u u u LR T   H U H (9) where u T U is a unimodular matrix with integer elements ( det | | 1u T  U ); ˆ u LR H is the channel matrix in the LR domain. The precoding matrix buW for the uth user is created by applying the ZF algorithm on ˆ u LR H . Finally, the precoding matrix bW for all users is expressed as follows: 1 2 0 0 0 0 . 0 0 T R b b KN Nb b K                W W W W       (10) It can be seen that the LC-RBD-LR-ZF precoder involves numerous QR decomposition operations. Besides, the size of the matrices aW and bW increases linearly with the number of users. Therefore, this precoder is suitable for small size MU-MIMO systems. For massive MIMO systems with large number of antennas at the BS to serve dozens of users, the complexity of the LC-RBD-LR-ZF precoder becomes so high that it could hardly be applicable. B. Element-based Lattice Reduction (ELR) Algorithm The ELR algorithm was proposed by Qi Zhou and Xiaoli Ma in [20]. The Journal of Science and Technique - N.205 (3-2020) - Le Quy Don Technical University 61 algorithm aims at minimizing the elements on the main diagonal of the error covariance matrix, which is defined as [20]: 1( ) ,H C H H (11) where A BN NH  . As shown in [20], the ELR algorithm gives better performance than the SA and LLL lattice reduction algorithms. Moreover, the computational complexity of the ELR algorithm is significantly reduced compared to those of the SA and LLL ones. Therefore, the ELR algorithm is a suitable candidate for large MIMO systems. The ELR algorithm has two versions: 1) element-base lattice reduction shortest longest basis (ELR-SLB); and 2) the element-base lattice reduction shortest longest vector (ELR-SLV). Among the two, the ELR-SLB algorithm minimizes all elements on the diagonal of C . The algorithm completes when all the diagonal elements of C are irreducible. On the contrary, the ELR-SLV algorithm selects the largest element on the diagonal of C to reduce. The algorithm is finished when the largest element on the diagonal of C is irreducible. To balance the computational complexity and system performance, in this paper, we adopt the ELR-SLB algorithm as a part of our proposed precoder. For convenience, the ELR-SLB algorithm is summarized in Algorithm 1. Algorithm 1 The ELR-SLB algorithm 1. Input , , A BN NA BN N H  2. Compute 1( )H C H H and set BN  T I 3. Do: 4. Find the largest element , .k kC 5. Compute , i k  6. Compute 2 * * , , , , , , ,i k i k i i i k i k i k i kC C C         and chooses index 1, , ,arg max .Ri N i k i ki    7. If: , 0 , [1, ]i k Ai k N    go to step 12 8. ' ' ',k k i k it t t  9. ,k k i k ic c c    10. *, k k i i kc c c    11. While (true): 12. Output: 1( )HT T , and LR H HT Algorithm 2 The ZF-GP-LR precoding algorithm 1. Input , ,T R orrcN N H 2. Decide the number of user groups L and compute the size of the sub-matrices. 3. Generate the matrix 1corrH 4. Apply QR decomposition to 1extH 5. Generate the matrix 1 a GPW 6. Repeat Step 3 to Step 5 for the next user group until the precoding matrices l a GPW are obtained for all user groups. 7. Generate the matrix aGPW as in (14). 8. Generate the matrix 1 1 1 a corr GPH H W 9. Convert 1( ) TH into 1LRH by utilizing Algorithm 1. 10. Create the matrix 1 b ZFW 11. Repeat Step 8 to Step 10 for the next user group until the precoding matrices l b GPW are obtained for all user groups. 12. Generate the matrix bGPW as in (22). 13. Output: ,GP corr W Journal of Science and Technique - N.205 (3-2020) - Le Quy Don Technical University 62 4. Proposed ZF-GP-LR precoder A. Proposed ZF-GP-LR precoder In this section, based on the method in [16], we present a linear group precoding method in combination with the low complexity ELR-SLB technique for massive MIMO systems using the exponential correlation channel model. Block diagram of the proposed ZF-GP-LR precoder is described in Fig. 1. The overall precoding matrix for all users is defined as follows: ,a bcorr GP GP GPW W W (12) where ( )T TN LNaGP W  is designed to minimize the interferences from other user groups and ( )T RLN NbGP W  is designed to enhance the system performance. group division ... ... ... x Quantize x. . . . . . . . . n y Fig. 1. Block diagram of the proposed ZF-GP-LR precoder In the first step, the correlation channel matrix, corrH is divided into /( )RL L N  groups (i.e., sub-matrices) ( 1,2,..., )TNlcorr l L   H  where γ is an integer greater than one. The first group, 1corrH , consists of the first row to the γth row of the channel matrix corrH ; the second group, 2corrH , is from the (γ + 1)th row to the 2γth row; and the last group, LcorrH , is from the ( )RN  th row to the RN th row. Specifically, the correlation channel matrix from BS to all users can be represented as follows: 11 12 1 1 1 2 ( 1)1 ( 1)2 ( 1) ( )1 ( )2 ( ) 1 2 . T T T R R R T R R R T N corr N N corr N N N N L corr N N N N                                         h h h H h h h h h h H h h h H h h h                  (13) Journal of Science and Technique - N.205 (3-2020) - Le Quy Don Technical University 63 In the second step, the precoding matrix aGPW is designed to have the following form 1 2 ... , L a a a a GP GP GP GP   W W W W (14) where l a GPW is the precoding matrix for the lth group. To obtain l a GPW , let us first construct the channel matrix ( ) ( )R TN Nl corr  H  consisting of the channel coefficients for all groups except those for the lth group as the following 1 1 1( ) ...( ) ( ) ...( ) . Tl T l T l T L T corr corr corr corr corr     H H H H H (15) After that an extension of lcorrH is constructed as follows: { , }, l l l ext N corrH I H  (16) where ( ) ( ) ,R R TN N Nlext l RN N        H  and 2 .R n s N E    Applying QRD to lextH , we get ,lext l lH Q R (17) where ( ) ( )l T l TN N N Nl   Q  is an unitary matrix and lR is an upper triangular matrix. From lQ the precoding matrix aGPlW for the lth group is constructed as ( 1: , 1: ). l a GP l l l T l l TN N N N N N    W Q (18) After getting all the weight matrices , ( 1, , ) l a GP l LW  , we define the effective channel matrix for the lth group as follows: . l l a l corr GPH H W (19) The channel matrix ( )lH in (19) is then transposed and converted into the matrix LR lH in the LR domain by using the ELR-SLB algorithm to give ,LR Tl l lH U H  (20) where .TNLRl  H  The weight matrix l b ZFW for the lth group is created by applying the ZF procedure to LRlH as follows:      1 . l H Hb LR LR LR ZF l l l       W H H H   (21) Finally, the precoding matrix bGPW and unimodular matrix bGPU for all groups can be obtained as follows: Journal of Science and Technique - N.205 (3-2020) - Le Quy Don Technical University 64 1 2 1 2 0 0 0 0 0 0 0 0 , . 0 00 0 L b T ZF b T ZFb b GP GP Tb LZF                           W U W U W U UW             (22) In order to make sure that the transmit power is unchanged after the transmit signals are precoded, the normalized power factor GP is computed to be    .RGP Ha b a b GP GP GP GP N Tr        W W W W (23) The proposed algorithm ZF-GP-LR is summarized in Algorithm 2. At the user side, the received signal vector for all groups can be expressed as ( ) / .corr corr GP y H W x n (24) Using y in (25), the estimated signal vector is given by = x + 2 (25) where 111/ 2, (1 ), 2 RN z L m j R     1 is a column vector with RN ones, [ ]zQ a denotes th